https://github.com/saschatimme/linearcovariancemodels.jl

Compute MLEs for linear covariance models
https://github.com/saschatimme/linearcovariancemodels.jl

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Compute MLEs for linear covariance models

• Host: GitHub
• URL: https://github.com/saschatimme/linearcovariancemodels.jl
• Owner: saschatimme
• License: mit
• Created: 2019-07-26T15:23:41.000Z (over 5 years ago)
• Default Branch: master
• Last Pushed: 2021-08-20T13:27:39.000Z (over 3 years ago)
• Last Synced: 2024-11-11T00:39:29.048Z (about 2 months ago)
• Language: Julia
• Homepage: https://saschatimme.github.io/LinearCovarianceModels.jl/stable/
• Size: 169 KB
• Stars: 2
• Watchers: 3
• Forks: 2
• Open Issues: 0
• Metadata Files:
  • Readme: README.md
  • License: LICENSE

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README

        
# LinearCovarianceModels

[![][docs-stable-img]][docs-stable-url] [![Build Status](https://travis-ci.com/saschatimme/LinearCovarianceModels.jl.svg?branch=master)](https://travis-ci.com/saschatimme/LinearCovarianceModels.jl)

[`LinearCovarianceModels.jl`](https://github.com/saschatimme/LinearCovarianceModels) is a package for

computing Maximum Likelihood degrees and MLEs of linear covariance models using numerical nonlinear algebra.

In particular [HomotopyContinuation.jl](https://www.JuliaHomotopyContinuation.org).

## Installation

In order to use `LinearCovarianceModels.jl` you need to have at least Julia 1.1 installed. If this is not the case you can download it at [julialang.org](https://julialang.org). Please see the [platform specific instructions](https://julialang.org/downloads/platform.html) if you have trouble installing Julia.

The package can be installed by executing

```julia

julia> using Pkg; Pkg.add("LinearCovarianceModels")

```

in the Julia REPL.

If you are looking for a more IDE like experience take a look at [Juno](https://junolab.org).

## Introduction by Example

```julia

# load package

julia> using LinearCovarianceModels

# Create a linear covariance model

julia> Σ = toeplitz(3)

3-dimensional LCModel:

 θ₁  θ₂  θ₃

 θ₂  θ₁  θ₂

 θ₃  θ₂  θ₁

# Compute a witness for the ML degree

julia> W = ml_degree_witness(Σ)

 MLDegreeWitness:

 • ML degree → 3

 • model dimension → 3

 • dual → false

# We offer the option to numerically verify the ML Degree

julia> verify(W)

 Compute additional witnesses for completeness...

 Found 10 additional witnesses

 Found 10 additional witnesses

 Compute trace...

 Norm of trace: 2.6521474798326718e-12

 true

# Consider the sample covariance matrix S

julia> S = [4/5 -9/5 -1/25

            -9/5 79/16 25/24

            -1/25 25/24 17/16];

# We use the ML degree witness set W to compute all critical points of the MLE

# problem.

julia> critical_points(W, S)

3-element Array{Tuple{Array{Float64,1},Float64,Symbol},1}:

 ([2.39038, -0.286009, 0.949965], -5.421751313919751, :local_maximum)

 ([2.52783, -0.215929, -1.45229], -5.346601549034418, :global_maximum)

 ([2.28596, -0.256394, 0.422321], -5.424161999175718, :saddle_point)  

# If we are just interested in the MLE, there is also a shorthand.

julia> mle(W, S)

3-element Array{Float64,1}:

  2.527832268219689  

 -0.21592947057775033

 -1.4522862659134732

```

For more informations take a look at the [documentation](https://saschatimme.github.io/LinearCovarianceModels.jl/stable).

[docs-stable-img]: https://img.shields.io/badge/docs-stable-blue.svg

[docs-stable-url]: https://saschatimme.github.io/LinearCovarianceModels.jl/stable